Abstract
In this chapter we will generalize Proposition 3.4.6 of Hager about counting the zeros of holomorphic functions of exponential growth. In Hager and Sjöstrand (Math Ann 342(1):177–243, 2008. http://arxiv.org/abs/math/0601381) we obtained such a generalization, by weakening the regularity assumptions on the functions ϕ. However, due to some logarithmic losses, we were not quite able to recover Hager’s original result, and we still had a fixed domain Γ with smooth boundary.
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Notes
- 1.
Here “4” can be replaced by any fixed constant >2.
- 2.
In the following we simply write C instead of C new, following the general rule that constants change from time to time.
- 3.
That is, given by an equation f(z) = 0, where f belongs to a bounded family of smooth real functions with the property that f(z) = 0 ⇒ \(|\nabla f(z)|\ge 1/{\mathcal {O}}(1)\) uniformly for all f in the family.
- 4.
Which says that if \(\Omega \Subset \mathbf {C}\) is a connected open set with smooth boundary and K ⊂ Ω a compact subset, then there exists a constant C = C Ω,K > 0 such that u(z 1) ≤ Cu(z 2) for all z 1, z 2 ∈ K and every non-negative harmonic function u on Ω, uniformly if Ω varies in a family of uniformly bounded subsets of C and \(\mathrm {dist}\,(K,\partial \Omega )\ge 1/{\mathcal {O}}(1)\) uniformly, see e.g., [4, Section 6.3].
- 5.
This also follows more directly from the spherical mean value inequality for subharmonic functions.
References
L. Ahlfors, Complex analysis. An Introduction to the Theory of Analytic Functions of One Complex Variable, 3rd edn. International Series in Pure and Applied Mathematics (McGraw-Hill Book Co., New York, 1978)
P. Bleher, R. Mallison Jr., Zeros of sections of exponential sums. Int. Math. Res. Not. 2006, 1–49 (2006), Art. ID 38937
D. Borisov, P. Exner, Exponential splitting of bound states in a waveguide with a pair of distant windows. J. Phys. A 37(10), 3411–3428 (2004)
E.B. Davies, Eigenvalues of an elliptic system. Math. Z. 243(4), 719–743 (2003)
T. Fischer, Existence, Uniqueness, and Minimality of the Jordan Measure Decomposition. http://arxiv.org/abs/1206.5449
D. Grieser, D. Jerison, Asymptotics of the first nodal line of a convex domain. Invent. Math. 125(2), 197–219 (1996)
M. Hager, J. Sjöstrand, Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators. Math. Ann. 342(1), 177–243 (2008). http://arxiv.org/abs/math/0601381
B. Helffer, J. Sjöstrand, Multiple wells in the semiclassical limit. I. Commun. Partial Differ. Equ. 9(4), 337–408 (1984)
M. Hitrik, J. Sjöstrand, Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions IIIa. One branching point. Canad. J. Math. 60(3), 572–657 (2008)
B.Ya. Levin, Distribution of Zeros of Entire Functions, English translation (American Mathematical Society, Providence, 1980)
A. Pfluger, Die Wertverteilung und das Verhalten von Betrag und Argument einer speciellen Klasse analytischen Funktionen I, II. Comment. Math. Helv. 11, 180–214 (1938), 12, 25–65 (1939)
B. Shiffman, S. Zelditch, S. Zrebiec, Overcrowding and hole probabilities for random zeros on complex manifolds. Ind. Univ. Math. J. 57(5), 1977–1997 (2008)
J. Sjöstrand, Counting zeros of holomorphic functions of exponential growth. J. Pseudodiffer. Operators Appl. 1(1), 75–100 (2010). http://arxiv.org/abs/0910.0346
J. Sjöstrand, M. Vogel, Large bi-diagonal matrices and random perturbations. J. Spectr. Theory 6(4), 977–1020 (2016). http://arxiv.org/abs/1512.06076
M. Sodin, Zeros of Gaussian analytic functions. Math. Res. Lett. 7, 371–381 (2000)
M. Sodin, B. Tsirelson, Random complex zeroes. III. Decay of the hole probability. Isr. J. Math. 147, 371–379 (2005)
S. Zrebiec, The zeros of flat Gaussian random holomorphic functions on ℂn, and hole probability. Mich. Math. J. 55(2), 269–284 (2007)
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Sjöstrand, J. (2019). Counting Zeros of Holomorphic Functions. In: Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations. Pseudo-Differential Operators, vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10819-9_12
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